Communication
Effect of the Molecular Weight of AB
Diblock Copolymers on the Lamellar
Orientation in Thin Films: Theory and
Experiment
Igor I. Potemkin,* Peter Busch, Detlef-M. Smilgies, Dorthe Posselt,
Christine M. Papadakis
We propose a theoretical explanation of the parallel and perpendicular lamellar orientations
in free surface films of symmetric polystyrene-block-polybutadiene diblock copolymers on
silicon substrates (with a native SiOx layer). Two approaches are developed: A correction to the
strong segregation theory and a qualitative analysis
of the intermediate segregation regime. We show
that the perpendicular orientation of the lamellae
formed by the molecules of high molecular weight
is stabilized by A–B interfacial interactions. They
are weaker in the case of the perpendicular orientation of the lamellae, whereas the surface tension
coefficient of the A–B interface decreases with the
increase of the molecular weight.
I. I. Potemkin
Physics Department, Moscow State University, Moscow 119992,
Russian Federation
Department of Polymer Science, University of Ulm, 89069 Ulm,
Germany
E-mail: [email protected]
P. Busch
JCNS-FRMII, Technische Universität München, D-85747 Garching,
Germany
D.-M. Smilgies
Cornell High Energy Synchrotron Source (CHESS), Wilson
Laboratory, Cornell University, Ithaca, New York 14853, USA
D. Posselt
IMFUFA (Department of Mathematics and Physics), Roskilde
University, DK-4000 Roskilde, Denmark
C. M. Papadakis
Physics Department E13, Technische Universität München,
D-85747 Garching, Germany
Macromol. Rapid Commun. 2007, 28, 579–584
ß 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Introduction
Microphase separation of compositionally symmetric diblock copolymers in the bulk results in the formation of a
lamellar microstructure.[1,2] This bulk morphology is also
generated in relatively thin films of the diblock copolymers. However, the presence of two boundaries introduces
additional degrees of freedom. Most notable is the thermodynamic stability of two different orientations of the
lamellae: parallel and perpendicular with respect to the
boundaries. Crucial for many applications is the ability to
DOI: 10.1002/marc.200600764
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I. I. Potemkin, P. Busch, D.-M. Smilgies, D. Posselt, C. M. Papadakis
control the orientation.[3] It was shown in many experiments[4–7] and supported theoretically[8] that primarily a
difference in the affinity of the blocks toward the substrate(s) is responsible for the parallel orientation of the
lamellae, while similar interactions of both blocks with the
substrate(s) can promote a perpendicular orientation of
the lamellae.[9–12] Recently we have shown that in the case
of free surface films, the interaction parameters of the
blocks with the boundaries are not the only means to
control the lamellar orientation. Depending on the overall molecular weight, symmetric polystyrene-block-polybutadiene [PS-b-PB] diblock copolymers on silicon
substrates (with a native SiOx layer) reveal two distinct
orientations of the lamellae toward the substrate: parallel
(low molecular weight) and perpendicular (high molecular
weight).[13] In this paper we will present the orientation
diagram of the PS-b-PB film, which was obtained
using atomic force microscopy[13] and grazing-incidence
small-angle X-ray scattering (GISAXS),[16] and propose a
theoretical explanation of the observed phenomenon.
Experimental Part
The films were prepared from symmetric PS-b-PB diblock copolymers.[2,14] The volume fraction of PB in all the samples was
0.49 0.01. Nine copolymers with molar masses between 13.9 and
183 kg mol1 and with bulk lamellar thicknesses in the range
between 138 and 839 Å were used. The Flory–Huggins interaction
parameter is x ¼ A/T þ B with A ¼ (21.6 2.1)K and B ¼ 0.019 0.005.[14] In the bulk, we have identified the strong segregation
limit where the lamellar thickness scales with the overall number
of the statistical segments in the chain N as N0.61 and the lamellar
interfaces are sharp with nearly no mixing of different monomers,
to be reached only above xN 30.[2] Below this value, only
intermediate segregation is encountered, i.e. the concentration
profile is not rectangular, but rather sinusoidal, partial mixing of
the two types of monomers takes place, and the lamellar thickness
scales as N0.83.[14]
The films were prepared by dissolving the polymers in toluene
and spin-coating films onto Si wafers covered with a native silicon
oxide layer. The film thickness was controlled by varying
the concentration of the solutions (polymer concentration 0.1–
7 wt.-%) and was determined by spectroscopic ellipsometry. For
details of the film preparation see ref.[13] Samples with molar
masses 54.5 kg mol1 were annealed at 150 8C while samples
with molar masses below 54.5 kg mol1 were dried at room
temperature in order to avoid dewetting. The AFM images of the
films show a clear molar mass dependence of the lamellar orientation at the film surface: Low molar mass films (13.9–54.5 kg mol1) show terraces at the surface having a height similar to the
bulk lamellar thickness. High molar mass samples (148 and
183 kg mol1), on the other hand, show meandering surface
textures with repeat distances similar to the bulk lamellar
thickness. These results point to a parallel and a perpendicular
lamellar orientations near the film surface for low and high molar
masses, respectively. This behavior is independent of the reduced
580
Macromol. Rapid Commun. 2007, 28, 579–584
ß 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
film thickness in the range studied, i.e., the ratio between the
film thickness and the bulk lamellar thickness.[13] More insight
into the lamellar orientation in the interior of the films could
be gained using grazing-incidence small-angle X-ray scattering
(GISAXS[15–17]). Experiments were carried out at the synchrotron
beamlines ESRF ID 10B and at CHESS D-line. The incident angle of the
X-ray beam, ai, is chosen to be slightly above the critical angle of
external reflection of the polymer film, acP, which is 0.158 for PS-b-PB.
The incident angle is below the critical angle of the substrate. Under
these conditions, the X-rays penetrate the film completely, and
absorption by the substrate is minimized. Scattering in the plane of
incidence, i.e. with scattering vector normal to the film surface,
together with the scattering out of the plane of incidence, i.e. with
scattering vector parallel to the film surface, is recorded simultaneously by means of a two-dimensional detector. The resulting 2D
reciprocal space maps were analyzed in the framework of a model
based on the distorted-wave Born approximation (DWBA[18]).
The resulting lamellar orientations inside the films as determined
using GISAXS are compiled as a function of xN and the reduced
lamellar thickness, Dred, which is the ratio between the film
thickness and the bulk lamellar thickness, see Figure 1. Low molar
mass films have a parallel orientation (the only exception being
ultrathin films where no distinct orientation can be defined),
whereas two highest molar masses show a perpendicular orientation. Thick high molar mass films approach the bulk limit, i.e. in
addition to the perpendicular lamellar orientation, other orientations are observed. Below, we propose a theory explaining the
transition of the lamellar orientation as a function of molar mass.
Figure 1. Lamellar orientation in thin films of symmetric
polystyrene-block-polybutadiene diblock copolymers as observed
using GISAXS. The orientations are given in double-logarithmic
representation as a function of xN and of the reduced film
thickness, Dred, which is the ratio between the film thickness
and the bulk lamellar thickness. Triangles pointing up: parallel;
triangles pointing down: perpendicular; diamonds: coexisting
orientations. The solid and dotted curves correspond to the
theoretical estimates of the boundary between the orientations.
The arrow marks the value of the cross-over between intermediate and strong segregation regime identified for the P(S-b-B)
system in the bulk (ref.[2],[14]).
DOI: 10.1002/marc.200600764
Effect of the Molecular Weight of AB Diblock Copolymers . . .
Theoretical Part
An important feature determining the behavior of the free
surface films is the presence of only one confining solid
surface. Therefore, the thickness of the film with parallel
orientation of the lamellae is not a fixed parameter. It is
determined by the condition of thermodynamic equilibrium: if the amount of polymer in the film does not
conform to the optimum (equilibrium) periodicity of the
lamellae, the film is macroscopically separated into two
films (phases) differing in thickness but each having the
equilibrium period of the structure.
Theoretical predictions for the lamellar orientation in
the free surface films were formulated in ref.[19] in the
strong segregation approximation, xN 1. The strong
segregation theory is asymptotically exact in the limit
xN ! 1. The finite number of segments introduces corrections to the total free energy which can be taken into
account similar to the case of microphase separation in the
bulk.[20] Combining the approaches of ref.[19,20] we can
write the free energy of the symmetric structure of the
parallel lamellae (both boundary layers comprise the same
kind of blocks) in the following form:
fk ¼
pffiffiffi
pffiffiffi 2
Na xð1 xA Þ
16N xa
p2 Lk
þ
þ
ln
12 Na2
2Lk
pLk
(1)
Here the first term corresponds to the elastic free energy of
the blocks forming brush layers of thickness L||, in Figure 2,
a and N are the length and the number of the segments in
the chain, respectively. The second term is the energy of
the various interfaces.[19] The choice of the symmetric
structure in comparison with asymmetric (the boundary
layers are comprised of different kinds of blocks) is
done for determination and does not affect the general
pffiffiffi
conclusions of the calculations. In Equation (1), a x ¼ g 0 is
the main contribution to the surface tension coefficient of
the A–B interface, and xA is the ratio of the spreading
parameter of polymer A, SA, to g 0 and to the number of
layers, n þ 1, each of the thickness 2L||, xA ¼ SA/g 0(n þ 1).[19]
The spreading parameter SA is known to be a combination
of the surface tension coefficients of the substrate–air (g sa),
polymer–air (g Aa), and polymer–substrate (g As) interfaces,
SA ¼ g sa–g Aa–g As. The third term of Equation (1) contains
two contributions coming from the correction to the
surface tension coefficient of the A–B interface due to
localization of the junction points of the blocks at the
interface, and from the entropy of the chain ends.[20] The
equilibrium value of the free energy is found by minimization with respect to L|| which can be done using a
perturbation theory
fk f0 ð1 xA Þ
f0 ¼
2=3
þ ln
3pð1 xA Þ1=3
ð3pÞ2=3
ðxNÞ1=3 1
4
Lk L0 ð1 xA Þ1=3 1 þ
L0 ¼
64f0
3
p2
1=3
1
2f0 ð1 xA Þ2=3
ax1=6 N 2=3
!
;
!
;
ð2Þ
where f0 and L0 are the asymptotic values (xN ! 1) of the
free energy and of the layer thickness in the bulk,
respectively. Note that our model does not take into
account the difference in the stretching of the blocks in the
boundary layers and in the internal ones. This correction
has a smaller order of magnitude[21] than the logarithmic
term of Equation (2), and can be neglected.
The free energy of the film with perpendicular orientation
of the lamellae toward the substrate has a similar form:[19]
f? ¼
pffiffiffi
Na x NðSA þ SB Þ
p2 L2?
þ
2D
12 Na2
2L?
pffiffiffi 16N xa
þ ln
pL?
(3)
where D is the thickness of the film and
SB is the spreading parameter of the
blocks B.[19] Minimization of Equation (3)
gives
Figure 2. Schematic representation of parallel and perpendicular lamellar orientations in
thin films of symmetric diblock copolymers. L|| and L? denote the corresponding
thicknesses of the brush layers. It is assumed that each internal layer of A (gray) or
B (white) type of the parallel lamellae has thickness 2L|| (approximation of impenetrable
brush layers) whereas the boundary layers are half this thickness. The total number of
internal layers is denoted by n. The thickness of A and B layers of the perpendicular
lamellae is 2L?.
Macromol. Rapid Commun. 2007, 28, 579–584
ß 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
NðSA þ SB Þ
64f0
f? f0 þ ln
3p
2D
1
L? L0 1 þ
2f0
(4)
where the thickness of the brush layer L?
coincides with that in the bulk. A
condition for the transition between
www.mrc-journal.de
581
I. I. Potemkin, P. Busch, D.-M. Smilgies, D. Posselt, C. M. Papadakis
the parallel and perpendicular structures can be found by
equating the free energies, f|| ¼ f?, considering that the film
thickness D ¼ 2(n þ 1)L||:[19]
3 þ ða 2ÞxA 3ð1 xA Þ1=3
"
#
1
xA ða þ 1Þ
1=3
ð1 xA Þ lnð1 xA Þ þ
;
¼
f0
2ð1 xA Þ2=3
a¼
SB
SA
ð5Þ
k
This condition determines the relation between the
surface tension coefficients, the film thickness and the
length of the blocks at the transition. The phase diagram of
the film in the limit f0 ! 1 is presented in ref.[19] To fit the
experimental data, let us find the dependence of the
reduced film thickness (i.e. the ratio of the film thickness to
the period of the lamellar structure in the bulk), Dred, on N
at the condition given by Equation (5):
ðn þ 1ÞLk
D
¼
4L?
2L?
SA
pffiffiffi
2a x
Dred ð1 xA Þ1=3
xA
1
1
1þ
2f0 ð1 xA Þ2=3 2f0
!
(6)
This parameter as a function of f0 has the form J þ H/
f0þ . . . . Numerical calculations demonstrate that the
coefficient H is always negative which means that Dred
is an increasing function of N. A very simple expression for
Dred can be found for the case of thick enough films (or
small values of the spreading parameters), jxA j 1. In this
case the orientational transition occurs at
1þ
1
xA
b
; 1 a 1
f0 3ð1 aÞ
(7)
see Equation (5), and
Dred
SA
1
1
pffiffiffi
6a xð1 aÞ
f0
(8)
Equation (7) means that the perpendicular orientation of
the lamellae is stable at 1 þ 1/f0 < b
while the parallel
lamellae are formed at 1 þ 1/f0 > b
. Therefore, the
transition from the parallel to the perpendicular lamellae
can be induced by increasing the number of segments of
the chain, N( f0 N1/3).
To clarify the physical meaning of the lamellar reorientation upon variation of the molecular weight, let us
compare the different contributions to the total free
energies of both structures: the elastic energy, fel [the first
term of Equation (1) and (3)], A–B interfacial energy, fAB
582
[the first part of the second term of Equation (1) and the
second term of Equation (3)], the surface energy of the
boundaries of the film, fb [the second part of the second
term of Equation (1) and the third term of Equation (3)],
and the penalty in the free energy accounting for
localization of the junction points of the blocks at the
interfaces and the entropy of the chain ends, fje [the last
term of Equation (1) and (3)]
Macromol. Rapid Commun. 2007, 28, 579–584
ß 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
2xA
fel
;
1
fel?
3
k
fje
fje?
1þ
k
fAB
xA
? 1þ 3 ;
fAB
xA
;
3 lnð64f0 =3pÞ
jxA j ¼
k
fb
2
fb? 1 þ a
jxA j
1
ð1 þ 1=f0 Þ
(9)
Under wetting conditions (both spreading parameters are
k
k
k
k
?
> 1; fje =fje? > 1.
positive) fel =fel? < 1; fb =fb? < 1 and fAB =fAB
Therefore, we can say that the A–B interfacial interactions
and localization of the junction points stabilize the
perpendicular lamellar orientation. Increasing N ði:e: xA Þ
k
k
results in decreasing fel =fel? and fje =fje? but increasing
k
?
?
fAB =fAB
. Since the contribution fAB
, which is responsible for
the stability of the perpendicular structure, decreases on
k
increasing N (compared to fAB ), the A–B interfacial interactions are the driving forces for the reorientation. Indeed,
the dependence on N can be included in the effective
surface tension coefficient g ¼ g 0(1 þ 1/f0) which, when
decreasing, leads to the thinning of the parallel lamellae,
Lk =L? 1 xA =3, and to the increase in the A–B interfacial
k
area, i.e. of fAB .
As we mentioned above, the strong segregation regime
can be reached only above xN 30.[2,14] Below this value,
10 < xN < 30, only intermediate segregation is encountered. Indeed, in this range of values of xN, the main
correction to the strong segregation theory 1/f0 is not
negligible (for example, 1/f0 0.33 at xN ¼ 20) and the
perturbation theory has to be modified. It was predicted
theoretically[22–24] and confirmed experimentally[2,14,25]
that in the intermediate segregation regime, the domain
spacing in the bulk has a stronger dependence on N compared to the strong segregation regime. Let us assume that
in the intermediate segregation regime the stretching of
the blocks (the end-to-end distance) follows a power law,
Rd ax1=6 N 2=3 ðxNÞd ; d 0. Here the parameter d is not a
constant: With increasing xN, it gradually decreases from a
certain, positive value to 0. To calculate the free energy of
the lamellar structure in the bulk in the intermediate
segregation regime, let us refer to the blob picture. Each
chain can be considered as a sequence of blobs, each of size
j and with a number of segments g. The relation between j
and g is chosen from the condition that the chain is not
stretched at length scales smaller than j, i.e. j ag1/2. Then
the free energy of the chain is proportional to the number
DOI: 10.1002/marc.200600764
Effect of the Molecular Weight of AB Diblock Copolymers . . .
of the blobs fd N=g Rd =j ðxNÞ1=3þ2d . To apply the
results of the strong segregation theory to the case of the
intermediate segregation regime, let us substitute the real
chains by virtual ones having M segments of the size a.
We assume that the end-to-end distance of the virtual
chain coincides with the one of the real chain and follows
exactly the same power law as in the strong segregation
regime, Rd ax1=6 M2=3 . Here x differs from x to provide the
strong segregation condition. Equating the free energy of
the virtual chain to fd ; ðxMÞ1=3 fd , we get x xðxNÞ6d [or
g g 0 ðxNÞ3d for the surface tension coefficient]. Therefore, Dred of the system with the virtual chains at
the condition of the transition from the parallel to the
perpendicular lamellae [Equation (5) at f0 ! 1) has the
form
Dred SA ð1 xA Þ1=3
pffiffiffi
xA
2a x
Received: November 2, 2006; Accepted: December 6, 2006;
DOI: 10.1002/marc.200600764
(10)
i.e., Dred of the real system in the intermediate segregation
regime depends on N.
In order to fit the experimental data of Figure 1, let us
rewrite Equation (10) in the form
SA ð1 xA Þ1=3
¼ ln
pffiffiffi
2a x
xA
Acknowledgements: Financial support by NATO by a Collaborative Linkage Grant is gratefully acknowledged. I.I.P. thanks the
Deutsche Forschungsgemeinschaft within the SFB 569 and the
Russian Foundation for Basic Research for financial support. C.M.P.
and P.B. acknowledge financial support by Deutsche Forschungsgemeinschaft (Pa 771/1-1).
Keywords: block copolymers; films; grazing-incidence smallangle X-ray scattering (GISAXS); microstructure; theory
SA
ð1 xA Þ1=3
pffiffiffi
3d
xA
2a xðxNÞ
lnðDred Þ ¼ c 3d lnðxNÞ; c
perpendicular orientation of the lamellae in thin films can
be stable. The transition from the parallel to the perpendicular orientation with increasing molecular weight is driven by A–B interfacial interactions which are weaker for
longer molecules in the perpendicular orientation of the
lamellae.
!
(11)
where parameters c and d are considered as the fitting parameters. The dependence of d on xN can be found in
the limit xN 1 if we use the result for the surface tension coefficient in the strong segregation regime,[20]
g g 0 ½1 þ 0:9 lnðxNÞ=ðxNÞ1=3 ; d 0:3=ðxNÞ1=3 . Relying
on the experimental results for the parameter d in the
intermediate segregation regime,[14]d 0.22 at xN ¼ 25, we
choose c 2.5. The solid part of the boundary curve in
Figure 1 corresponds to Equation (11) with d 0.3/
(xN)1/3. The dotted part of the curve has an expected
monotonic behavior of the boundary whose d varies with
xN, and d 0.22 at the point ln(xN ¼ 25) 3.22 and
ln(Dred) 0.376. The physical reason for the lamellar
reorientation in the intermediate segregation regime
can also be found in the decrease of the effective surface
tension coefficient (g) with increasing N.
Conclusion
In conclusion, we demonstrate that depending on the
molecular weight of the compositionally symmetric
PS-b-PB diblock copolymers, both the parallel and the
Macromol. Rapid Commun. 2007, 28, 579–584
ß 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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DOI: 10.1002/marc.200600764